Lowpass filter

A lowpass filter passes signal frequencies near DC and attenuates high frequencies. The simplest first-order lowpass has transfer function

$H(s)=\frac{{\omega }_c}{s+{\omega }_c},H(j\omega )=\frac{{\omega }_c}{j\omega +{\omega }_c},$

where ${\omega }_c$ is the cutoff frequency in rad/s. (Filter transfer functions are usually written in ω-form; the formula sheet’s filter tables follow this convention.)

Behavior

• At DC ($\omega =0$): $H(0)={\omega }_c/{\omega }_c=1$. The signal passes unattenuated.
• As $\omega \to \infty$: $|H(j\omega )|\to 0$. High frequencies are blocked.
• At $\omega ={\omega }_c$: $|H(j{\omega }_c)|={\omega }_c/\sqrt{2{\omega }_c^2}=1/\sqrt{2}$. Magnitude has dropped to about $0.707$, which is a $3\text{dB}$ drop. So ${\omega }_c$ is the half-power cutoff.

This is the canonical first-order lowpass response. The RC lowpass we keep running into has exactly this form, with ${\omega }_c=1/(RC)$.

Pole-zero structure

One pole at $s=-{\omega }_c$ on the negative real axis. No zeros. The pole’s location in the left half-plane tells you the filter is stable; its distance from the imaginary axis (${\omega }_c$) determines how broadly the filter passes frequencies.

Bode plot

Asymptotic magnitude (in dB): flat at $0\text{dB}$ for $\omega \ll {\omega }_c$, then dropping at $-20\text{dB/decade}$ for $\omega \gg {\omega }_c$. The true curve is $-3\text{dB}$ at $\omega ={\omega }_c$, transitioning smoothly between the two asymptotes.

Phase: starts at $0°$ at low frequencies, transitions through $-45°$ at $\omega ={\omega }_c$, and asymptotes to $-90°$ at high frequencies. The transition spans two decades, from $\omega =0.1{\omega }_c$ to $\omega =10{\omega }_c$.

See Bode plot for the construction rules.

Impulse response

The inverse Laplace of ${\omega }_c/(s+{\omega }_c)$ is ${\omega }_c{e}^{-{\omega }_{c}t}u(t)$ — a decaying exponential with time constant $\tau =1/{\omega }_c$. So the filter’s “memory” persists for about $5\tau$ seconds.

Step response

The step response is the integral of the impulse response: $(1-e^{-{\omega }_{c}t})u(t)$. The familiar RC step response — exponential approach to a final value of 1.

Higher-order lowpasses

A second-order lowpass adds another pole, doubling the high-frequency rolloff to $-40\text{dB/decade}$. With complex-conjugate poles you can get a sharper transition near the cutoff at the cost of a peak (resonance) just before rolling off — see Resonance.

Order-$n$ lowpasses have $n$ poles total and roll off at $-20n\text{dB/decade}$ past cutoff. Higher order means sharper cutoff but more complex implementation and more phase distortion.

Where it’s used

• Audio: remove hiss above the audible band.
• Anti-aliasing before sampling — protects against high-frequency content aliasing into the band.
• Smoothing of measurement signals (sensor data, control feedback).
• “Integrator” approximation — over a band well above ${\omega }_c$, the magnitude is ${\omega }_c/\omega$, which is like a $1/\omega$ integrator.

Circuit realisation (Electronics I)

The concrete first-order lowpass in electronics is the RC lowpass filter: a series resistor feeding a shunt capacitor, output across the capacitor, with Cutoff frequency $f_c=1/(2\pi RC)$ and a $-20\text{dB/decade}$ rolloff above it. The capacitor is an open at DC (signal passes) and a short at high frequency (output grounded), which is the physical mechanism behind this transfer function.